So intervals overlap - That's why I can call a major 3rd a diminished 4th, and such.

However, from what I understand, a perfect 5th only overlaps with a diminished 6th. You cannot refer to a 7 semitone interval using a 4th because the "biggest" fourth, augmented fourth, is a diminished 5th.

So consider this (treble clef):

What would you call the interval here? From what I understand it should be called a 4th because of the distance between the notes. But that's impossible.

Welcome to Music! The answers are correct that these notes, written this way are a double-augmented forth. It's also worth noting that it's not impossible to write it in this way, but it's pretty unusual to mix sharps and flats in this way. They could also be expressed as G# + D# (a perfect fifth) or as Ab + Eb (also a perfect fifth). When listening, you can't differentiate between a double-augmented forth and a perfect fifth, you'll hear a perfect fifth, it's just "spelled" as a forth. Which one to use depends heavily on context, both key signature and implied key.
– AJFaradayMar 15 at 13:36

4 Answers
4

But since "augmented fourth" won't be big enough for this, we kind of had to make up a term, and the world of music theory collectively decided upon calling this interval a doubly augmented fourth. This just means that it's one half step larger than an augmented fourth. (As such, the augmented fourth is not really the "biggest" fourth possible.)

The same is true for diminished intervals; a half step smaller than a diminished sixth will be a doubly diminished sixth.

Any interval can overlap with (or, in more academic terms, "be enharmonic to") another interval. A perfect fifth could, in theory, be enharmonic to a triply augmented third—not that you'll encounter those very often!

What would one call the interval between B♯ and C♭? An "ascending" doubly-diminished second?
– supercatMar 14 at 15:08

5

@Tim In real music, these intervals are very rare. But in the extra credit portions of my exams, they suddenly appear all over the place :-)
– RichardMar 14 at 16:47

1

@Tim i'd say that's realistic, and a good way to really understand the relationships between intervals. I had a bunch of them on the exams I took. In the beginning it was annoying, but they were fun from the second quarter onward
– AethenosityMar 14 at 21:55

2

@Tim These are also for extra credit as a way to test the really excellent students. The normal questions don't involve these crazy intervals.
– RichardMar 14 at 21:59

1) It's a fourth because there are 4 letters from the lower note to the higher (A, B (or H if you're using German note names, just like me), C, and D).So, A<some accidental> to D<some accidental> will always be a 4th, regardless of the actual accidentals or their absence.

2) Since A to D (without any accidentals) is a perfect 4th, and this must be 2 half-steps wider than A-D (since A&flat; is 1 half-step lower than A, and D&sharp; is 1 half-step higher than D), so A&flat; to D&sharp; would be a doubly augmented 4th (because we augment the perfect interval twice).

And another 2 pence following on from Richard's and trolley813's answers:

1) First, you name the interval according to the note names. So an A​<something> up to a D​<something> will always be a fourth of some kind. (This is the diatonic bit, based on how the interval looks.)

2) Then, you adjust it so it covers the right number of semitones. (This is the chromatic bit, based on how the interval sounds.)

That second step is always possible, as you can squash or stretch any numbered interval as much as you need, by diminishing or augmenting.

I've had two professors at different universities who used the terms "super" and "hyper" used instead of "doubly" or "double" to refer to intervals which are a semitone (half step) larger than augmented—or smaller than diminished.

I am not asserting that this is "correct" or "standard" or even a "known & accepted variation"; however, it is something which you (or anyone) may encounter at some point. One was not a native English speaker and both were trained in non-English speaking schools/countries; I assume that is a factor.
I think most people would be able to figure this out, but someone inevitably asked what the heck they were talking about every time I heard them use the term—although, in those contexts I suspect the same question would have been asked if they used the standard terminology.

I argued the point to one of them that "hyper diminished" is actually incorrect and makes not sense at all. I asserted that if he insisted on using substandard jargon, he should at least use it correctly and say "hypo diminished". This started a drawn out discussion among most of the class which was eventually "settled" by an English professor (confirming my position).

This reminds me of a little game I used to play with a high school buddy of mine: "What would be the key signature of…?" (followed by some ridiculous key like: B double-sharp minor) We used to say that we should have unique, dedicated words for "double-flat" and "double-sharp", as well as "doubly diminished" and "doubly augmented".

Apparently, there are several languages which had those words but they have fallen out of use—I believe German or Russian was one of those languages.

In Russian, we don't have (and didn't in the past) separate words for double sharp and double flat, however the "double-" part in these words is borrowed from French (instead of native Russian counterpart), so it makes them somewhat separate. On the other hand, doubly/triply/etc diminished and augmented intervals use native Russian word for the multiplicity (doubly, triply)
– trolley813Mar 18 at 7:10